Why was everything unlocked the first time I started Portal 2 coop?
This is actually a feature. Here's Portal 2 developer PersonGuy's explanation:
Because the game is running 'offline', it doesn't have access to the progress data in Steam. To make this work we'll need to rewrite quite a bit splitscreen session code that assumes multiple log ins.
So instead of making you start at the very beginning every session, Valve just unlocks all the levels for local co-op.
As for the intro cut-scene and level, those may be accessed by pausing the game in the level select chamber and selecting the menu item for "Calibration course."